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-
1. Let \(U_1,U_2,U_3\leq \R ^3\) be given by:
\(\seteqnumber{0}{}{0}\)
\begin{align*}
U_1&=\Span {(1,0,-1)};\\ U_2&=\Span {(1,1,1), (1,-1,0)};\\ U_3&=\set {x\in \R ^3\st x_2+x_3=0}.
\end{align*}
Which of the three sums \(U_i+U_j\), \(1\leq i<j\leq 3\), are direct? In each case, briefly justify your answer. [4]
-
2. Let \(U=\Span {(0,1,2)}\leq \R ^3\). Define subspaces \(W_1,W_2\leq \R ^3/U\) by
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
W_1=\Span {(1,1,1)+U},\qquad W_2=\Span {(2,4,6)+U}.
\end{equation*}
Is \(W_1+W_{2}\) direct? Justify your answer. [4]
-
3. Let \(p=(x-7)^2(x-3)\in \R [x]\) and let \(A\) be a square matrix such that \(p(A)=0\).
What are the possibilities for the minimum polynomial of \(A\)?
How does your answer change if you know that \(A\) is not diagonal? [4]
-
6.
-
(a) Let \(U\) be a linear subspace of a possibly infinite-dimensional vector space \(V\).
Suppose that the quotient space \(V/U\) is finite-dimensional. Show that \(U\) has a complement in \(V\). [6]
-
(b) Let \(A\) be given by
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
\begin{pmatrix*}[r] 2&-1&0\\6&-3&-2\\-6&3&4 \end {pmatrix*}.
\end{equation*}
[9]
-
7.
-
(a) Let \(B\in M_3(\R )\) be given by
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
\begin{pmatrix*}[r] 1&-1&0\\0&-2&1\\1&-3&1 \end {pmatrix*}
\end{equation*}
-
(i) Show that the minimum polynomial of \(B\) is \(x^{3}\).
-
(ii) Can the following basis of \(\R ^3\) be re-ordered to give a Jordan basis for \(B\)? If so, how? If not, find a Jordan basis for \(B\).
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
(0,0,1),\qquad (0,1,1), \qquad (2,2,4).
\end{equation*}
[10]
-
(b) Contemplate the symmetric bilinear form \(B_A\) on \(\R ^5\) where
\(\seteqnumber{0}{}{0}\)
\begin{equation*}
A= \begin{pmatrix*}[r] 1&2&3&4&5\\2&-1&6&7&8\\3&6&0&9&10\\4&7&9&-3&11\\5&8&10&11&4 \end {pmatrix*}.
\end{equation*}
Let \((p,q)\) be the signature of \(B_A\). Show that \(p,q>0\). [5]